$\text{ If the second term in the expansion}{\left(\sqrt[\text{'}13]{a}\text{+}\frac{a}{\sqrt{a^{-1}}}\right)}^n\text{is } 14a^{\frac{5}{2}}\text{ then the value of }\frac{{}^{n}C_3}{{}^{n}C_2}\text{ is }$

$$\begin{gathered} \text{ We have, }{T}_2=14a^{\frac{5}{2}} \\ \Rightarrow {}^{n}C_1{\left(a^{\frac{1}{13}}\right)}^{n-1}{\left(a^{\frac{3}{2}}\right)}^1=14a^{\frac{5}{2}} \\ \Rightarrow n{a}^{\frac{n-1}{13}+\frac{3}{2}}=14{\mathrm{a}}^{\frac{5}{2}} \\ \Rightarrow \mathrm{n}=14 \\ \Rightarrow \frac{{}^{\mathrm{n}}\mathrm{C}_3}{{}^{\mathrm{n}}\mathrm{C}_2}=\frac{{}^{14}\mathrm{C}_3}{{}^{14}\mathrm{C}_2}=\frac{12}{3}=4 \end{gathered}$$